Description
Statistics for Political Science
August 7, 2025
Norms of the Class
- Focus on getting better, not on being “good.”
- This program is your job.
This class is designed to give you the skills to pursue your own independent research projects in the future; don’t worry about becoming perfect.
Where are we? Where are we going?
- Today: Descriptive Statistics
- Thursday: Data Manipulation
- Friday: Data Visualization
Today’s Agenda
“Correlation doesn’t equal causation”
…but all our statistical evidence about causation is built on correlations.
We need to understand the basics of statistical correlations:
- Mean and median
- Variance and standard deviation
- Covariance and correlation
- Visualizing statistical relationships
A Note on Notation
We will introduce quite a lot of notation in the next few weeks. We have tried to keep notation to a minimum, but it is often necessary to communicate complex ideas quickly and precisely.
Notation today:
- \(Y\) = typically the dependent variable.
- \(n\) = the total number of observations in sample \(n\).
- \(i\) = an individual within the sample \(n\).
- \(Y_i\) = the value of variable \(Y\) for individual \(i\).
- \(\sum_{i=1}^n Y_i\) = the sum of all values of \(Y\) for each individual \(i\) in sample \(n\).
- \(\bar{Y}\) (“Y-bar”) = mean of sample \(Y\).
Why Description Matters
Quantitative political science seeks to understand political phenomena through numerical representation.
- Description is the foundation of all statistical work.
- It tells us: What can we say about the data we have?
- Before inference or prediction, we need to understand our data.
Example: Weeks For Abortion
“Different states are debating when, if at all, abortion should be legal during a woman’s pregnancy. A normal pregnancy could go up to as many as 40 weeks. Until what point in a pregnancy do you think a woman should be legally allowed to obtain an abortion?”
Respondents choose number of weeks (0–40)
Can we just look at the data?
What about a frequency table?
What if we want to compare by party affiliation?
What if we want to compare by age?
Why do we need statistical description?
Fundamental tension in quantitative analysis: detail vs parsimony.
- Sample Mean 17.08 weeks.
- Sample Median: 15 weeks.
Compare averages across categories:
- (Mean \(\mid\) Republican) = 10.5
- (Mean \(\mid\) Democrat) = 22.8
Summarize relationships across multiple continuous variables:
- Correlation between Age and Abortion Weeks = –0.086
Central Tendency
Central Tendency
Central tendency refers to measures that identify the center of a dataset.
Continuous and ordinal variables:
- Mean: average value.
- Median: value at ≥50% of the distribution.
- Mode: more frequently occurring value.¹
Categorical variables: report as tables or recode in binary.
¹ This value is often meaningless for continuous variables, so is rarely included.
Mean
\[ \bar{Y} = \frac{\sum_{i=1}^n Y_i}{n} \]
Components:
- \(\bar{Y}\) (“Y-bar”) = mean of \(Y\)
- \(\sum_{i=1}^n Y_i\) = the sum of all values of \(Y\)
- \(n\) = the total number of observations
Properties of Mean
1. Zero-Sum Property
\[ \sum_{i=1}^n (Y_i - \bar{Y}) = 0 \]
2. Least-Squares Property
\[
\sum_{i=1}^n (Y_i - \bar{Y})^2 \;<\;
\sum_{i=1}^n (Y_i - c)^2
\quad \forall\; c \neq \bar{Y}
\]
Median
The median is the middle value of a variable when the observations are ordered from smallest to largest.
It divides the distribution into two equal halves: 50% of values are below the median and 50% are above it.
If there is an odd number of observations, the median is the middle value.
If there is an even number of observations, the median is the average of the two middle values.
Key advantage: The median is resistant to outliers and skewed data.
Age
Age: Mean is 52.48
Age: Median is 56
Binary
Binary: Mean is 0.623
Binary: Median is 1
Mean vs. Median
Imagine there are 5 people sitting in a bar:
- Net worth: $10,000, $120,000, $140,000, $250,000
- Mean: $130,000
- Median: $130,000
Imagine that Elon Musk walks into the bar:
- Net worth: $10,000, $120,000, $140,000, $250,000, $393,000,000,000
- Mean: $78,600,104,000
- Median: $140,000
Dispersion
Variance
Variance is the typical (squared) distance between an observation and the mean:
\[ \mathrm{var}(Y) = s^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n} \]
- Calculate each observation’s distance from the mean: \((Y_i - \bar{Y})\)
- Square the distances: \((Y_i - \bar{Y})^2\)
- Now they’re all positive.
- Bigger differences matter more.
- Now they’re all positive.
- Take the average of the squared differences.
Standard Deviation
\[ \mathrm{sd}(Y) = s = \sqrt{\frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}} \]
Assuming a normal distribution:
- 68% of observations will be within one standard deviation.
- 95% of observations will be within two standard deviations.
- 99.5% of observations will be within three standard deviations.
Age
Age: One Standard Deviation
Age: Two Standard Deviations
Age: Three Standard Deviations
10 Minute Break
Bivariate Description
Covariance
\[ \mathrm{cov}(X, Y) = \frac{\sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y})}{n} \]
- If \(X\) and \(Y\) tend to increase together, most terms are positive → positive covariance.
- If there’s no clear relationship, positives and negatives cancel → covariance ≈ 0.
cov(Age, Weeks of Abortion) = -18.4
Correlation
\[ \mathrm{cor}(X,Y) = \frac{\mathrm{cov}(X,Y)}{\mathrm{sd}(X)\,\mathrm{sd}(Y)} \]
- \((\mathrm{cor}(X,Y)=1)\) → perfect positive linear relationship.
- \((\mathrm{cor}(X,Y)=-1)\) → perfect negative linear relationship.
- \((\mathrm{cor}(X,Y)=0)\) → no linear relationship.
cor(Age, Weeks of Abortion) = -0.086
Correlation
Data Visualization
Why Visualize Data?
- Helps identify patterns and outliers
- Makes data easier to interpret
- Communicates findings effectively
Histogram
Histograms?
Histograms allow us to visualize the distribution of a single continuous variable.
- The x-axis displays the bins or intervals of the variable.
- The y-axis shows how many observations fall into each bin.
Box Plots
Box Plots?
Box plots help us compare the distributions of a continuous variable across categories.
- The continuous variable is on the y-axis.
- The categorical variable is on the x-axis.
Scatter Plots
Scatter Plots?
We use scatter plots to visualize the relationship between two continuous variables:
- The independent variable is on the x-axis.
- The dependent variable is on the y-axis.